Instruments such as the piano have notes-- an A is 440hz, for example. While it may not sound like a piano, the tone() function lets us generate these sounds. How would I (or could I) generate more "percussion" or "atonal" noises, such as drums of the noise of stomping your foot? What frequency are these? Can I generate them?

I don't know many options so far, as I'm new to Arduino. But, I did manage to get multiple PCM sounds from a single Arduino chip. Here's a great article about how to generate PCM sounds with the Arduino.